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Efficient Algorithms for the Longest Path Problem

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Title: Efficient Algorithms for the Longest Path Problem


1
Efficient Algorithms for the Longest Path Problem
  • Ryuhei UEHARA (JAIST)
  • Yushi UNO (Osaka Prefecture University)

2
The Longest Path Problem
  • Finding a longest (vertex disjoint) path in a
    given graph
  • Motivation (comparing to Hamiltonian path)
  • … Approx. Algorithm, Parameterized Complexity
  • … More practical/natural
  • … More difficult(?)

3
The Longest Path Problem
  • Known (hardness) results
  • We cannot find a path of length n-ne in a given
    Hamiltonian graph in poly-time unless PNP
    Karger, Motwani, Ramkumar 1997
  • We can find O(log n) length path Alon, Yuster,
    Zwick1995
  • (?O((log n/loglog n)2) Björklund, Husfeldt
    2003)
  • Approx. Alg. achieves O(n/log n) AYZ95
  • (?O(n(loglog n/log n)2)BH03)
  • Exponential algorithm Monien 1985

4
The Longest Path Problem
  • Known polynomial time algorithm
  • Dijkstras Alg.(196?)Linear alg. for finding a
    longest path in a tree

5
The Longest Path Problem
  • Known polynomial time algorithm
  • Dijkstras Alg.(196?)Linear alg. for finding a
    longest path in a tree

6
The Longest Path Problem
  • Known polynomial time algorithm
  • Dijkstras Alg.(196?)Linear alg. for finding a
    longest path in a tree

7
The Longest Path Problem
  • Known polynomial time algorithm
  • Dijkstras Alg.(196?)Linear alg. for finding a
    longest path in a tree

8
Approaches to the Efficient Algs to Longest Path
Problem
  • Extension of the Dijkstras algorithm
  • Weighted trees (linear), block graphs (linear),
    cacti (O(n2)).
  • Graph classes s.t. Hamiltonian Path can be found
    in poly time
  • Some graph classes having interval
    representations
  • (bipartite permutation, interval biconvex
    graphs)
  • Dynamic programming to the graph classes that
    have tree representations (on going)
  • Cacti(linear), …

(ISAAC 2004)
(ISAAC 2004)
9
Approaches to the Efficient Algs to Longest Path
Problem
  • Extension of the Dijkstras algorithm
  • Weighted trees (linear), block graphs (linear),
    cacti (O(n2)).
  • Graph classes s.t. Hamiltonian Path can be found
    in poly time
  • Some graph classes having interval
    representations
  • (bipartite permutation, interval biconvex
    graphs)
  • Dynamic programming to the graph classes that
    have tree representations (on going)
  • Cacti(linear), …

(ISAAC 2004)
(ISAAC 2004)
10
1. Ex of Dijkstras Alg
  • Bulterman et.al. (IPL,2002) showed that
  • the correctness of Dijkstras alg stands for
  • For each u,v,
  • length of the shortest path between u and v
  • length of the longest path between u and v
  • For each u,v,w,
  • d(u,v) ? d(u,w) d(w,v)
  • For each u,v,w,
  • d(u,v) d(u,w) d(w,v) if and only if
  • w is on the unique path between u and v

11
1. Ex of Dijkstras Alg
  • Construct G(V,E) from G(V,E) s.t.
  • V?V
  • For each u,v?V,
  • length of the shortest path between u,v on G
  • length of the longest path between u,v on
    G
  • For each u,v?V,
  • the shortest path between u,v on G is unique

12
1. Ex of Dijkstras Alg
  • Theorem ExDijkstra finds a longest path if G and
    G satisfy the conditions.
  • ExDijkstra G(V,E) and G(V,E)
  • pick any vertex w in V
  • find x?V with maxd(w,x) on G
  • find y?V with maxd(x,y) on G
  • x and y are the endpoints of the longest path in
    G, and d(x,y) on G is its length.

13
1. Ex of Dijkstras Alg (Summary)
  • Theorem Vertex/edge weighted tree (linear)
  • Theorem Block graph (O(VE))
  • Theorem Cactus (O(V2))

14
1. Ex of Dijkstras Alg (Cacti)
  • Cactus
  • Each block is a cycle
  • Two cycle share at most one vertex which is a
    separator

G
G
cactus
The longest path between u and v on G
The shortest path between u and v on G

15
1. Ex of Dijkstras Alg (Cacti)
  • Sample

16
1. Ex of Dijkstras Alg (Cacti)
  • Sample

17
1. Ex of Dijkstras Alg (Cacti)
  • Sample

18
1. Ex of Dijkstras Alg (Cacti)
  • Sample

19
1. Ex of Dijkstras Alg (Cacti)
  • Sample

20
1. Ex of Dijkstras Alg (Cacti)
  • Sample

21
1. Ex of Dijkstras Alg (Cacti)
  • Sample

22
Graph classes s.t. Hamiltonian Path can be found
in poly time
  • Fact 1
  • Hamiltonian Path is NP-hard on a chordal graph.
    (In fact, strongly chordal split
    graphMüller,1997.)
  • Fact 2
  • Hamiltonian Path is solvable on an interval graph
    in linear time. Damaschke, 1993.
  • Our goal
  • Poly-time algorithm for Longest Path on an
    interval graph.

23
Interval Graphs
  • An interval graph G(V,E) has an interval
    representation s.t. u,v?E iff IunIv?f

24
Interval Graphs
  • An interval graph G(V,E) has an interval
    representation s.t. u,v?E iff IunIv?f

Iv
v
Iu
Hamiltonian Path linear time solvable.
u
Longest Path ????
? Restricted interval graphs…
25
Restricted Interval Graphs
  • An interval biconvex graph G(S?Y,E) has an
    interval representation s.t…

Y biconvex
s1
s3
s6
s8
s2
s4
s7
s5
S integer points
26
Restricted Interval Graphs
  • Interval biconvex graph G(S?Y,E) is introduced
    Uehara, Uno 2004 from graph theoretical
    viewpoints
  • Natural analogy of biconvex graphs (bipartite
    graph class)
  • Generalization of proper interval graphs
  • Generalization of threshold graphs
  • Best possible class longest path can be found in
    poly time…

27
Poly-time alg for longest path on an interval
biconvex graph (idea)
  • Find the trivial longest path P on GY
  • Embed the vertices in S into P as possible
  • Adjust endpoints if necessary.

3
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3
7
1
9
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30
28
Poly-time alg for longest path on an interval
biconvex graph (idea)
  • Find the trivial longest path P on GY
  • Embed the vertices in S into P as possible
  • Adjust endpoints if necessary.

3
4
3
7
1
9
2
3
8
8
30
29
Poly-time alg for longest path on an interval
biconvex graph (idea)
  • Find the trivial longest path P on GY
  • Embed the vertices in S into P as possible
  • Adjust endpoints if necessary.

3
4
7
9
8
8
30
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3
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1
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2
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Poly-time alg for longest path on an interval
biconvex graph (idea)
  • Find the trivial longest path P on GY
  • Embed the vertices in S into P as possible
  • Adjust endpoints if necessary.

3
4
7
9
8
30
30
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7
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31
Poly-time alg for longest path on an interval
biconvex graph (idea)
  • Find the trivial longest path P on GY
  • Embed the vertices in S into P as possible
  • Adjust endpoints if necessary.

3
  • How can we determine the vertices in S?
  • Where do we embed them?

4
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32
Poly-time alg for longest path on an interval
biconvex graph (idea)
  • Embed the vertices in S into P as possible

w(e) is the number at the right-endpoint
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33
Poly-time alg for longest path on an interval
biconvex graph (idea)
  • Embed the vertices in S into P as possible

w(e) is the number at the right-endpoint
3
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7
1
9
2
3
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8
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30
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Find the maximum weighted matching!!
34
Open Problems
  • Longest Path on an interval graph??
  • Combination of DP/Dijkstra and weighted maximum
    matching on MPQ-tree representation?
  • Related to the following open problem?
  • Hamiltonian Path with a start point on an
    interval graph? Damaschke, 1993.
  • Extension to
  • Longest cycle on some graph classes
  • Hamiltonian cycle/path on some graph classes
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